Theorem iotass 4904

Description: Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.)

Assertion

Ref	Expression
iotass	⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → (℩𝑥𝜑) ⊆ 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem iotass

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iota 4887	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
2		unieq 3610	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦})
3		vex 2604	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	3	unisn 3617	. . . . . . . 8 ⊢ ∪ {𝑦} = 𝑦
5	2, 4	syl6eq 2129	. . . . . . 7 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = 𝑦)
6		df-pw 3384	. . . . . . . . . . 11 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	6	sseq2i 3024	. . . . . . . . . 10 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴 ↔ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴})
8		ss2ab 3062	. . . . . . . . . 10 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴} ↔ ∀𝑥(𝜑 → 𝑥 ⊆ 𝐴))
9	7, 8	bitri 182	. . . . . . . . 9 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ⊆ 𝐴))
10	9	biimpri 131	. . . . . . . 8 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → {𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴)
11		sspwuni 3760	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∣ 𝜑} ⊆ 𝐴)
12	10, 11	sylib 120	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → ∪ {𝑥 ∣ 𝜑} ⊆ 𝐴)
13		sseq1 3020	. . . . . . . 8 ⊢ (∪ {𝑥 ∣ 𝜑} = 𝑦 → (∪ {𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
14	13	biimpa 290	. . . . . . 7 ⊢ ((∪ {𝑥 ∣ 𝜑} = 𝑦 ∧ ∪ {𝑥 ∣ 𝜑} ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
15	5, 12, 14	syl2anr 284	. . . . . 6 ⊢ ((∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) ∧ {𝑥 ∣ 𝜑} = {𝑦}) → 𝑦 ⊆ 𝐴)
16	15	ex 113	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → ({𝑥 ∣ 𝜑} = {𝑦} → 𝑦 ⊆ 𝐴))
17	16	ss2abdv 3067	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ {𝑦 ∣ 𝑦 ⊆ 𝐴})
18		df-pw 3384	. . . 4 ⊢ 𝒫 𝐴 = {𝑦 ∣ 𝑦 ⊆ 𝐴}
19	17, 18	syl6sseqr 3046	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝒫 𝐴)
20		sspwuni 3760	. . 3 ⊢ ({𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝒫 𝐴 ↔ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝐴)
21	19, 20	sylib 120	. 2 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝐴)
22	1, 21	syl5eqss 3043	1 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → (℩𝑥𝜑) ⊆ 𝐴)

Syntax hints:   → wi 4  ∀wal 1282   = wceq 1284  {cab 2067   ⊆ wss 2973  𝒫 cpw 3382  {csn 3398  ∪ cuni 3601  ℩cio 4885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887

This theorem is referenced by:  fvss  5209  riotaexg  5492